The cases in proving that group of order 90 is not simple

  • Context: Undergrad 
  • Thread starter Thread starter Mr Davis 97
  • Start date Start date
  • Tags Tags
    Group
Click For Summary
SUMMARY

The discussion focuses on proving that the group of order 90 is not simple by analyzing two distinct cases. The first case involves counting elements to derive a contradiction, while the second case addresses the intersection of Sylow 3-subgroups. The reasoning is grounded in Lagrange's theorem, which asserts that the intersection of subgroups must also be a subgroup, leading to the conclusion that two distinct Sylow 3-subgroups cannot intersect trivially. This analysis is supported by class notes referenced on pages 65-66.

PREREQUISITES
  • Understanding of group theory concepts, particularly Sylow theorems
  • Familiarity with Lagrange's theorem in the context of subgroup intersections
  • Knowledge of group orders and their implications on subgroup structures
  • Ability to analyze mathematical proofs and contradictions
NEXT STEPS
  • Study the Sylow theorems in detail to understand their applications in group theory
  • Review Lagrange's theorem and its implications for subgroup intersections
  • Examine examples of non-simple groups and their properties
  • Explore advanced group theory texts for deeper insights into group orders and structure
USEFUL FOR

Mathematicians, particularly those specializing in group theory, students studying abstract algebra, and anyone interested in the properties of finite groups.

Mr Davis 97
Messages
1,461
Reaction score
44
https://imgur.com/a/FuCPJLe

I am trying to attempt this problem, but I am wondering why exactly these are the two cases the problem is split into. I can understand the first case, since that let's us count elements and get a contradiction, but why is the second case there? In other words, why do these two cases exhaust all possibilities?

EDIT: Actually, I think that I see it now. Since the intersection of subgroups is a subgroup, by Lagrange we must have that the negation of two distinct Sylow 3-subgroups intersecting trivially is intersecting with 3 elements.
 
Last edited:
Physics news on Phys.org
see pages 65-66 of these class notes:

http://alpha.math.uga.edu/%7Eroy/843-1.pdf
 

Similar threads

Undergrad Is G simple?Is G simple? A different approach using Sylow theorems
  • · Replies 5 ·
Replies
5
Views
3K
Undergrad Differences between left vs right actions in some group theory questions
  • · Replies 1 ·
Replies
1
Views
1K
Undergrad Subgroup axioms for a symmetric group
  • · Replies 1 ·
Replies
1
Views
1K
Undergrad Determining the number of conjugacy classes in a p-group
  • · Replies 6 ·
Replies
6
Views
3K
Insights Groups, The Path from a Simple Concept to Mysterious Results
  • · Replies 17 ·
Replies
17
Views
10K
Undergrad Question about "A Group Epimorphism is Surjective"
  • · Replies 13 ·
Replies
13
Views
2K
Undergrad Why only normal subgroup is used to obtain group quotient
  • · Replies 5 ·
Replies
5
Views
3K
Undergrad Dfns group action & group, written in terms of the other
  • · Replies 26 ·
Replies
26
Views
1K
Graduate Proving the Simplicity of a Group of Order 36 Using Sylow Subgroups
  • · Replies 9 ·
Replies
9
Views
9K
Undergrad First Sylow Theorem: Group of Order ##p^k## & Cyclic Groups
  • · Replies 1 ·
Replies
1
Views
2K